Journal of South China University of Technology (Natural Science Edition) ›› 2010, Vol. 38 ›› Issue (6): 95-99.doi: 10.3969/j.issn.1000-565X.2010.06.018

• Computer Science & Technology • Previous Articles     Next Articles

Lower Bounds of Second-Order Nonlinearity of Boolean Functions

Li Xue-lian1  Hu Yu-pu2  Gao Jun-tao2   

  1. 1.Department of Applied Mathematics,Xidian University,Xi＇ an 710071,Shannxi,China;2.School of Telecommunications Engineering,Xidian University,Xi＇ an 710071,Shannxi,China
  • Received:2009-08-11 Revised:2010-01-22 Online:2010-06-25 Published:2010-06-25
  • Contact: 李雪莲（1979-），女，讲师，博士生，主要从事密码函数、流密码研究． E-mail:xuelian202@163．com
  • About author:李雪莲（1979-），女，讲师，博士生，主要从事密码函数、流密码研究．
  • Supported by:

    国家“973”计划项目（2007CB311201）; 国家自然科学基金资助项目（60833008 60803149）; 广西信息与通讯技术重点实验室资助项目（20902）

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Abstract:

This paper deals with the second-order nonlinearities of the Boolean functions f（x）=tr（∑（n-1）/2」i,j=1bijxd） with n variables,where d=2i＋2j＋1,bij GF（2） and 1≤ij≤L（n-1）/2」.The derivatives with the maximal nonlinearity of f（x） are determined for odd n,and,for even n,the derivatives which are semi-Bent functions are obtained.Based on these derivatives with high nonlinerity,the tight lower bounds of the second-order nonlinearity of f（x） are given.The results show that f（x） with high second-order nonlinearity,can resist the quadratic and affine approximation attacks.

Key words: Boolean functions, cryptography, nonlinearities, Walsh transforms, derivatives